Abstract Wallach DMS-9531908 This project will continue the principal investigator's research in the representation theory of reductive groups and its applications to number theory, geometry and physics. One of the main themes in this project involves the study of (so called) singular unitary representations using a technique we call transfer which "moves" unitary representations between real forms of the same complexification. To implement this idea one must have very detailed information on the restriction of representations to (generally non-compact) symmetric subgroups. This analysis is of interest in its own right and has led (and will lead) researchers (including the PI) to surprising generalizations of Howe's theory of reductive dual pairs to exceptional groups. It will also lead to confirmation of special cases of some remarkable conjectures of B. Gross and D. Prasad. Another key direction of this research involves the continued study of the module theory of the ring of invariant polynomial differential operators on a reductive Lie algebra. This work has already led to a new interpretation of the Springer correspondence. The new direction here will involve some sort of "relative Springer correspondence". This project also involves joint work with Matthew Clegg on the development of a symbolic algebra package (to be called "Groebner") designed for large scale computations using parallel supercomputers or distributed networks of work stations. The solution to the problems described in this project would expand our understanding of some of the most exciting developments in the interaction of representation theory with apparently unrelated parts of mathematics and physics. For example the detailed analysis of one singular representation (by Kostant and his coworkers) has led to new solutions to Einstein's equations. Representation theory of the type studied in this project is also an important ingredient in the Langland's program of which one very small part due to Wiles led to his (Wile's) proof of Fermat's Last Theorem.
